DOCSLIB.ORG

Singular Value Inequalities for Hilbert Space Operators

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Singular Value Inequalities for Hilbert Space Operators Filomat 32:8 (2018), 2861–2866 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL1808861T University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat Singular Value inequalities for Hilbert space Operators Mahdi Taleb Alfakhra, Mohsen Erfanian Omidvara aDepartment of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran. Abstract. In this paper we show that if Ai; Bi; Xi are Hilbert space operators such that Xi is compact i = 1; 2;:::; n and f , 1 are non-negative continuous functions on [0; ) with f (t)1(t) = t for all t [0; ), also 1 2 1 h is non-negative increasing operator convex function on [0; ), then 1 0 0 11 0 1 Xn Xn Xn B B CC B 2 2 C h Bsj B !iAi∗Xi∗BiCC sj B !ih(Ai∗ f ( Xi∗ ) Ai) !ih(Bi∗1( Xi ) Bi)C @ @ AA ≤ @ j j ⊕ j j A i=1 i=1 i=1 Pn for j = 1; 2;::: and i=1 !i = 1. Also, applications of some inequalities are given. 1. Introduction Let be a complex Hilbert space with inner product ; and let ( ) denote the C∗-algebra of all boundedH linear operators on a complex separable Hilbert spaceh· ·i . ForB a compactH operator X ( ), let H 1 2 B H s (X) > s (X) > denote the singular values of X; i.e., the eigenvalues of X = (X∗X) 2 (The absolute value 1 2 ··· j j of X), arranged in decreasing order are repeated according to multiplicity. Note that sj(X) = sj(X∗) = sj( X ) for j = 1; 2;:::. For A; B ( ), we utilize the direct sum notation A B for the block-diagonal operatorj j " # 2 B H ⊕ A 0 defined on . It has been shown in [8] that if X and Y are compact operators, then 0 B H ⊕ H sj(X + Y) 6 2sj(X Y) ⊕ for j = 1; 2;:::. The usual operator norm of an operator A ( ) is denote by A = sup Ax : x = 1 . As an immediate consequence of the min-max principle2 (see B H e.g.,[2, p. 75]), ifjjA;jjB; and Xfjjarejj in jj( jj ) suchg that X is compact, then B H sj(AXB) 6 A B sj(X) (1) jj jj jj jj 1 p p for j = 1; 2;:::. For 1 6 p < , the Schatten p-norm of compact operator A is defined by A p = (tr A ) , where tr is the usual trace functional.1 One can show that jj jj j j A B = max( A ; B ) jj ⊕ jj jj jj jj jj 2010 Mathematics Subject Classification. Primary 47A30; Secondary 15A18, 47A63, 47TB10. Keywords. singular value, unitarily Invariant norm, positive operator Received: 21 June 2017; Accepted: 26 December 2017 Communicated by Fuad Kittaneh Email addresses: [email protected] (Mahdi Taleb Alfakhr), [email protected] (Mohsen Erfanian Omidvar) M. Taleb Alfakhr, M. E. Omidvar / Filomat 32:8 (2018), 2861–2866 2862 and p p 1 A B = ( A + B ) p : jj ⊕ jjp jj jjp jj jjp In addition to the usual operator norm, which is defined on all of ( ), we consider unitarily invariant norms . Each of these norms is defined on a norm ideal containedB inH the ideal of compact operators, and for thejjj sake·jjj of brevity, we will make no explicit mention of this ideal. Thus, when we consider X , we are assuming that the operator X belongs to the norm ideal associated with . Moreover, eachjjj unitarilyjjj invariant norm is symmetric gauge function of the singular values, andjjj· isjjj characterized by equality X = UXV forjjj·jjj all operator X and for all unitary operators U and V ( ). For the general theory of unitarilyjjj jjj jjj invariantjjj norms, we refer to [2] or [7]. It has been shown by Bhatia2 B andH Kittaneh in [4] that if A; B are compact operators in ( ), then B H 2 A∗B 6 AA∗ + BB∗ (2) jjj jjj jjj jjj and A∗B + B∗A 6 AA∗ + BB∗ (3) jjj jjj jjj jjj for every unitarily invariant norm. The inequality (2) has attracted the attention of several mathematicians, and different proof of a stronger version of it have been given. See [3], [8], [9] and [11]. It has been shown by Kittaneh [10] the generalized from of the mixed Schwarz inequality, that if A is an operator in ( ) and f and 1 are non-negative functions on [0; ) which are continuous and satisfy the relation f (t)1B(t)H= t for all t [0; ), then 1 2 1 Ax; y f ( A∗ )x 1( A )y jh ij ≤ j j j j for all x and y in . In this paper, weH generalize inequalities (2) and (3) and present a bound that involves operator A and B. 2. Main Results In this section, we establish generalized singular value inequalities for Hilbert space operator. The following Lemmas are essential role in our analysis. The first one on the Mixed schwarz inequality has been proven by Kittaneh [10]. " # AC Lemma 2.1. Let A; B; C ( ), such that A and B are positive, then T = ∗ is a positive in B( ) if 2 B H CB H ⊕ H and only if Cx; y 2 6 Ax; x By; y for all x; y in . jh ij h ih i H We prove the second one by Lemma 2.1. Lemma 2.2. Let A; B and X be operators in ( ) and let f and 1 be non-negative continuous functions on [0; ) that satisfy the relation f (t)1(t) = t for all t B[0;H ), then 1 2 1 2 2 3 6 A∗ f ( X∗ ) AA∗X∗B 7 6 j j 7 6 7 46 2 57 B∗XA B∗1( X ) B j j is positive. Proof. For any x; y in H 2 1 2 1 B∗XAx; y = XAx; By 6 f ( X∗ ) Ax; Ax 2 1( X ) By; By 2 : jh ij jh ij h j j i h j j i By Lemma 2.1 2 2 3 6 A∗ f ( X∗ ) AA∗X∗B 7 6 j j 7 6 7 > 0: 46 2 57 B∗XA B∗1( X ) B j j M. Taleb Alfakhr, M. E. Omidvar / Filomat 32:8 (2018), 2861–2866 2863 The third Lemma is Theorem 2.1 in [1]. " # AC∗ Lemma 2.3. Let A, B and C be compact operators in ( ) and is positive. Then sj(C) 6 sj(A B) for B H CB ⊕ j = 1; 2;:::. Now we establish a general singular value inequality,from which singular value inequalities for products of operators follow as special cases. Theorem 2.4. Let Ai,Bi and Xi be operators in ( ), where Xi is compact operator i = 1; 2;:::; n and let f and 1 be non-negative functions on [0; ), which are continuousB H and satisfy the relation f (t)1(t) = t for all t [0; ), also let h be a non-negative increasing1 operator convex function on [0; ). Then 2 1 0 0 11 0 1 1 Xn Xn Xn B B CC B 2 2 C h Bsj B !iA∗X∗BiCC sj B !ih(A∗ f ( X∗ ) Ai) !ih(B∗1( Xi ) Bi)C @B @B i i ACAC ≤ @B i j i j ⊕ i j j AC i=1 i=1 i=1 Pn for j = 1; 2;::: and positive real numbers !i such that i=1 !i = 1. Proof. The matrix for i = 1; 2;:::; n 2 2 3 6 Ai∗ f ( Xi∗ ) Ai Ai∗Xi∗Bi 7 6 j j 7 6 7 > 0 (by Lemma 2.2). 46 2 57 B∗XiAi B∗1( Xi ) Bi i i j j So, by Lemma 2.3 and some property of singular values, we have 0 1 0 1 Xn Xn Xn B C B 2 2 C sj B A∗X∗BiC 6 sj B A∗ f ( X∗ ) Ai B∗1( Xi ) BiC : (4) @B i i AC @B i j i j ⊕ i j j AC i=1 i=1 i=1 for j = 1; 2;:::. Now consider the non-negative increasing operator convex function h on [0; ) and in the 1 inequality (4), put p!iAi, p!iBi instead of Ai, Bi respectively . It follows that 0 0 11 0 0 11 Xn Xn Xn B B CC B B 2 2 CC h Bsj B !iA∗X∗BiCC 6 h Bsj B !iA∗ f ( X∗ ) Ai !iB∗1( Xi ) BiCC @B @B i i ACAC @B @B i j i j ⊕ i j j ACAC i=1 i=1 i=1 0 0 11 Xn B B 2 2 CC = h Bsj B !i(A∗ f ( X∗ ) Ai B∗1( Xi ) Bi)CC @B @B i j i j ⊕ i j j ACAC i=1 0 0 11 Xn B B 2 2 CC = sj Bh B !i(A∗ f ( X∗ ) Ai B∗1( Xi ) Bi)CC @B @B i j i j ⊕ i j j ACAC i=1 (by elementary functional calculus) 0 1 Xn B 2 2 C 6 sj B !ih(A∗ f ( X∗ ) Ai B∗1( Xi ) Bi)C @B i j i j ⊕ i j j AC i=1 (h is operator convex) 0 1 Xn Xn B 2 2 C = sj B !ih(A∗ f ( X∗ ) Ai) !ih(B∗1( Xi ) Bi)C @B i j i j ⊕ i j j AC i=1 i=1 for j = 1; 2;:::. Corollary 2.5. Let Ai; Bi; Xi ( ) such that Xi is compact for i = 1; 2;:::; n.
Load more

Recommended publications
  • Singular Value Decomposition (SVD)
    San José State University Math 253: Mathematical Methods for Data Visualization Lecture 5: Singular Value Decomposition (SVD) Dr. Guangliang Chen Outline • Matrix SVD Singular Value Decomposition (SVD) Introduction We have seen that symmetric matrices are always (orthogonally) diagonalizable. That is, for any symmetric matrix A ∈ Rn×n, there exist an orthogonal matrix Q = [q1 ... qn] and a diagonal matrix Λ = diag(λ1, . , λn), both real and square, such that A = QΛQT . We have pointed out that λi’s are the eigenvalues of A and qi’s the corresponding eigenvectors (which are orthogonal to each other and have unit norm). Thus, such a factorization is called the eigendecomposition of A, also called the spectral decomposition of A. What about general rectangular matrices? Dr. Guangliang Chen | Mathematics & Statistics, San José State University3/22 Singular Value Decomposition (SVD) Existence of the SVD for general matrices Theorem: For any matrix X ∈ Rn×d, there exist two orthogonal matrices U ∈ Rn×n, V ∈ Rd×d and a nonnegative, “diagonal” matrix Σ ∈ Rn×d (of the same size as X) such that T Xn×d = Un×nΣn×dVd×d. Remark. This is called the Singular Value Decomposition (SVD) of X: • The diagonals of Σ are called the singular values of X (often sorted in decreasing order). • The columns of U are called the left singular vectors of X. • The columns of V are called the right singular vectors of X. Dr. Guangliang Chen | Mathematics & Statistics, San José State University4/22 Singular Value Decomposition (SVD) * * b * b (n>d) b b b * b = * * = b b b * (n<d) * b * * b b Dr.
    [Show full text]
  • Eigen Values and Vectors Matrices and Eigen Vectors
    EIGEN VALUES AND VECTORS MATRICES AND EIGEN VECTORS 2 3 1 11 × = [2 1] [3] [ 5 ] 2 3 3 12 3 × = = 4 × [2 1] [2] [ 8 ] [2] • Scale 3 6 2 × = [2] [4] 2 3 6 24 6 × = = 4 × [2 1] [4] [16] [4] 2 EIGEN VECTOR - PROPERTIES • Eigen vectors can only be found for square matrices • Not every square matrix has eigen vectors. • Given an n x n matrix that does have eigenvectors, there are n of them for example, given a 3 x 3 matrix, there are 3 eigenvectors. • Even if we scale the vector by some amount, we still get the same multiple 3 EIGEN VECTOR - PROPERTIES • Even if we scale the vector by some amount, we still get the same multiple • Because all you’re doing is making it longer, not changing its direction. • All the eigenvectors of a matrix are perpendicular or orthogonal. • This means you can express the data in terms of these perpendicular eigenvectors. • Also, when we find eigenvectors we usually normalize them to length one. 4 EIGEN VALUES - PROPERTIES • Eigenvalues are closely related to eigenvectors. • These scale the eigenvectors • eigenvalues and eigenvectors always come in pairs. 2 3 6 24 6 × = = 4 × [2 1] [4] [16] [4] 5 SPECTRAL THEOREM Theorem: If A ∈ ℝm×n is symmetric matrix (meaning AT = A), then, there exist real numbers (the eigenvalues) λ1, …, λn and orthogonal, non-zero real vectors ϕ1, ϕ2, …, ϕn (the eigenvectors) such that for each i = 1,2,…, n : Aϕi = λiϕi 6 EXAMPLE 30 28 A = [28 30] From spectral theorem: Aϕ = λϕ 7 EXAMPLE 30 28 A = [28 30] From spectral theorem: Aϕ = λϕ ⟹ Aϕ − λIϕ = 0 (A − λI)ϕ = 0 30 − λ 28 = 0 ⟹ λ = 58 and
    [Show full text]
  • Singular Value Decomposition (SVD) 2 1.1 Singular Vectors
    Contents 1 Singular Value Decomposition (SVD) 2 1.1 Singular Vectors . .3 1.2 Singular Value Decomposition (SVD) . .7 1.3 Best Rank k Approximations . .8 1.4 Power Method for Computing the Singular Value Decomposition . 11 1.5 Applications of Singular Value Decomposition . 16 1.5.1 Principal Component Analysis . 16 1.5.2 Clustering a Mixture of Spherical Gaussians . 16 1.5.3 An Application of SVD to a Discrete Optimization Problem . 22 1.5.4 SVD as a Compression Algorithm . 24 1.5.5 Spectral Decomposition . 24 1.5.6 Singular Vectors and ranking documents . 25 1.6 Bibliographic Notes . 27 1.7 Exercises . 28 1 1 Singular Value Decomposition (SVD) The singular value decomposition of a matrix A is the factorization of A into the product of three matrices A = UDV T where the columns of U and V are orthonormal and the matrix D is diagonal with positive real entries. The SVD is useful in many tasks. Here we mention some examples. First, in many applications, the data matrix A is close to a matrix of low rank and it is useful to find a low rank matrix which is a good approximation to the data matrix . We will show that from the singular value decomposition of A, we can get the matrix B of rank k which best approximates A; in fact we can do this for every k. Also, singular value decomposition is defined for all matrices (rectangular or square) unlike the more commonly used spectral decomposition in Linear Algebra. The reader familiar with eigenvectors and eigenvalues (we do not assume familiarity here) will also realize that we need conditions on the matrix to ensure orthogonality of eigenvectors.
    [Show full text]
  • A Singularly Valuable Decomposition: the SVD of a Matrix Dan Kalman
    A Singularly Valuable Decomposition: The SVD of a Matrix Dan Kalman Dan Kalman is an assistant professor at American University in Washington, DC. From 1985 to 1993 he worked as an applied mathematician in the aerospace industry. It was during that period that he first learned about the SVD and its applications. He is very happy to be teaching again and is avidly following all the discussions and presentations about the teaching of linear algebra. Every teacher of linear algebra should be familiar with the matrix singular value deco~??positiolz(or SVD). It has interesting and attractive algebraic properties, and conveys important geometrical and theoretical insights about linear transformations. The close connection between the SVD and the well-known theo1-j~of diagonalization for sylnmetric matrices makes the topic immediately accessible to linear algebra teachers and, indeed, a natural extension of what these teachers already know. At the same time, the SVD has fundamental importance in several different applications of linear algebra. Gilbert Strang was aware of these facts when he introduced the SVD in his now classical text [22, p. 1421, obselving that "it is not nearly as famous as it should be." Golub and Van Loan ascribe a central significance to the SVD in their defini- tive explication of numerical matrix methods [8, p, xivl, stating that "perhaps the most recurring theme in the book is the practical and theoretical value" of the SVD. Additional evidence of the SVD's significance is its central role in a number of re- cent papers in :Matlgenzatics ivlagazine and the Atnericalz Mathematical ilironthly; for example, [2, 3, 17, 231.
    [Show full text]
  • Limited Memory Block Krylov Subspace Optimization for Computing Dominant Singular Value Decompositions
    Limited Memory Block Krylov Subspace Optimization for Computing Dominant Singular Value Decompositions Xin Liu∗ Zaiwen Weny Yin Zhangz March 22, 2012 Abstract In many data-intensive applications, the use of principal component analysis (PCA) and other related techniques is ubiquitous for dimension reduction, data mining or other transformational purposes. Such transformations often require efficiently, reliably and accurately computing dominant singular value de- compositions (SVDs) of large unstructured matrices. In this paper, we propose and study a subspace optimization technique to significantly accelerate the classic simultaneous iteration method. We analyze the convergence of the proposed algorithm, and numerically compare it with several state-of-the-art SVD solvers under the MATLAB environment. Extensive computational results show that on a wide range of large unstructured matrices, the proposed algorithm can often provide improved efficiency or robustness over existing algorithms. Keywords. subspace optimization, dominant singular value decomposition, Krylov subspace, eigen- value decomposition 1 Introduction Singular value decomposition (SVD) is a fundamental and enormously useful tool in matrix computations, such as determining the pseudo-inverse, the range or null space, or the rank of a matrix, solving regular or total least squares data fitting problems, or computing low-rank approximations to a matrix, just to mention a few. The need for computing SVDs also frequently arises from diverse fields in statistics, signal processing, data mining or compression, and from various dimension-reduction models of large-scale dynamic systems. Usually, instead of acquiring all the singular values and vectors of a matrix, it suffices to compute a set of dominant (i.e., the largest) singular values and their corresponding singular vectors in order to obtain the most valuable and relevant information about the underlying dataset or system.
    [Show full text]
  • Chapter 6 the Singular Value Decomposition Ax=B Version of 11 April 2019
    Matrix Methods for Computational Modeling and Data Analytics Virginia Tech Spring 2019 · Mark Embree [email protected] Chapter 6 The Singular Value Decomposition Ax=b version of 11 April 2019 The singular value decomposition (SVD) is among the most important and widely applicable matrix factorizations. It provides a natural way to untangle a matrix into its four fundamental subspaces, and reveals the relative importance of each direction within those subspaces. Thus the singular value decomposition is a vital tool for analyzing data, and it provides a slick way to understand (and prove) many fundamental results in matrix theory. It is the perfect tool for solving least squares problems, and provides the best way to approximate a matrix with one of lower rank. These notes construct the SVD in various forms, then describe a few of its most compelling applications. 6.1 Eigenvalues and eigenvectors of symmetric matrices To derive the singular value decomposition of a general (rectangu- lar) matrix A IR m n, we shall rely on several special properties of 2 ⇥ the square, symmetric matrix ATA. While this course assumes you are well acquainted with eigenvalues and eigenvectors, we will re- call some fundamental concepts, especially pertaining to symmetric matrices. 6.1.1 A passing nod to complex numbers Recall that even if a matrix has real number entries, it could have eigenvalues that are complex numbers; the corresponding eigenvec- tors will also have complex entries. Consider, for example, the matrix 0 1 S = − . " 10# 73 To find the eigenvalues of S, form the characteristic polynomial l 1 det(lI S)=det = l2 + 1.
    [Show full text]
  • Lecture 15: July 22, 2013 1 Expander Mixing Lemma 2 Singular Value
    REU 2013: Apprentice Program Summer 2013 Lecture 15: July 22, 2013 Madhur Tulsiani Scribe: David Kim 1 Expander Mixing Lemma Let G = (V; E) be d-regular, such that its adjacency matrix A has eigenvalues µ1 = d ≥ µ2 ≥ · · · ≥ µn. Let S; T ⊆ V , not necessarily disjoint, and E(S; T ) = fs 2 S; t 2 T : fs; tg 2 Eg. Question: How close is jE(S; T )j to what is \expected"? Exercise 1.1 Given G as above, if one picks a random pair i; j 2 V (G), what is P[i; j are adjacent]? dn #good cases #edges 2 d Answer: P[fi; jg 2 E] = = n = n = #all cases 2 2 n − 1 Since for S; T ⊆ V , the number of possible pairs of vertices from S and T is jSj · jT j, we \expect" d jSj · jT j · n−1 out of those pairs to have edges. d p Exercise 1.2 * jjE(S; T )j − jSj · jT j · n j ≤ µ jSj · jT j Note that the smaller the value of µ, the closer jE(S; T )j is to what is \expected", hence expander graphs are sometimes called pseudorandom graphs for µ small. We will now show a generalization of the Spectral Theorem, with a generalized notion of eigenvalues to matrices that are not necessarily square. 2 Singular Value Decomposition n×n ∗ ∗ Recall that by the Spectral Theorem, given a normal matrix A 2 C (A A = AA ), we have n ∗ X ∗ A = UDU = λiuiui i=1 where U is unitary, D is diagonal, and we have a decomposition of A into a nice orthonormal basis, m×n consisting of the columns of U.
    [Show full text]
  • Singular Value Decomposition
    Math 312, Spring 2014 Jerry L. Kazdan Singular Value Decomposition Positive Definite Matrices Let C be an n × n positive definite (symmetric) matrix and consider the quadratic polynomial hx;Cxi. How can we understand what this\looks like"? One useful approach is to view the image 2 2 of the unit sphere, that is, the points that satisfy kxk = 1. For instance, if hx;Cxi = 4x1 + 9x2 , 2 2 then the image of the unit disk, x1 + x2 = 1, is an ellipse. For the more general case of hx;Cxi it is fundamental to use coordinates that are adapted to the matrix C . Since C is a symmetric matrix, there are orthonormal vectors v1 , v2 ,. , vn in which C is a diagonal matrix. These vectors vj are eigenvectors of C with corresponding eigenvalues λj : so Cvj = λjvj . In these coordinates, say x = y1v1 + y2v2 + ··· + ynvn: (1) and Cx = λ1y1v1 + λ2y2v2 + ··· + λnynvn: (2) Then, using the orthonormality of the vj , 2 2 2 2 kxk = hx; xi = y1 + y2 + ··· + yn and 2 2 2 hx;Cxi = λ1y1 + λ2y2 + ··· + λnyn: (3) For use in the singular value decomposition below, it is con- ventional to number the eigenvalues in decreasing order: λmax = λ1 ≥ λ2 ≥ · · · ≥ λn = λmin; so on the unit sphere kxk = 1 λmin ≤ hx;Cxi ≤ λmax: As in the figure, the image of the unit sphere is an \ellipsoid" whose axes are in the direction of the eigenvectors and the corresponding eigenvalues give half the length of the axes. There are two closely related approaches to using that a self-adjoint matrix C can be orthogonally diagonalized.
    [Show full text]
  • A High Performance Block Eigensolver for Nuclear Configuration Interaction Calculations Hasan Metin Aktulga, Md
    1 A High Performance Block Eigensolver for Nuclear Configuration Interaction Calculations Hasan Metin Aktulga, Md. Afibuzzaman, Samuel Williams, Aydın Buluc¸, Meiyue Shao, Chao Yang, Esmond G. Ng, Pieter Maris, James P. Vary Abstract—As on-node parallelism increases and the performance gap between the processor and the memory system widens, achieving high performance in large-scale scientific applications requires an architecture-aware design of algorithms and solvers. We focus on the eigenvalue problem arising in nuclear Configuration Interaction (CI) calculations, where a few extreme eigenpairs of a sparse symmetric matrix are needed. We consider a block iterative eigensolver whose main computational kernels are the multiplication of a sparse matrix with multiple vectors (SpMM), and tall-skinny matrix operations. We present techniques to significantly improve the SpMM and the transpose operation SpMMT by using the compressed sparse blocks (CSB) format. We achieve 3–4× speedup on the requisite operations over good implementations with the commonly used compressed sparse row (CSR) format. We develop a performance model that allows us to correctly estimate the performance of our SpMM kernel implementations, and we identify cache bandwidth as a potential performance bottleneck beyond DRAM. We also analyze and optimize the performance of LOBPCG kernels (inner product and linear combinations on multiple vectors) and show up to 15× speedup over using high performance BLAS libraries for these operations. The resulting high performance LOBPCG solver achieves 1.4× to 1.8× speedup over the existing Lanczos solver on a series of CI computations on high-end multicore architectures (Intel Xeons). We also analyze the performance of our techniques on an Intel Xeon Phi Knights Corner (KNC) processor.
    [Show full text]
  • Lecture 6: October 19, 2015 1 Singular Value Decomposition
    Mathematical Toolkit Autumn 2015 Lecture 6: October 19, 2015 Lecturer: Madhur Tulsiani 1 Singular Value Decomposition Let V; W be finite-dimensional inner product spaces and let ' : V ! W be a linear transformation. Since the domain and range of ' are different, we cannot analyze it in terms of eigenvectors. However, we can use the spectral theorem to analyze the operators '∗' : V ! V and ''∗ : W ! W and use their eigenvectors to derive a nice decomposition of '. This is known as the singular value decomposition (SVD) of '. Proposition 1.1 Let ' : V ! W be a linear transformation. Then '∗' : V ! V and ''∗ : W ! W are positive semidefinite linear operators with the same non-zero eigenvalues. In fact, we can notice the following from the proof of the above proposition Proposition 1.2 Let v be an eigenvector of '∗' with eigenvalue λ 6= 0. Then '(v) is an eigen- vector of ''∗ with eigenvalue λ. Similarly, if w is an eigenvector of ''∗ with eigenvalue λ 6= 0, then '∗(w) is an eigenvector of '∗' with eigenvalue λ. Using the above, we get the following 2 2 2 ∗ Proposition 1.3 Let σ1 ≥ σ2 ≥ · · · ≥ σr > 0 be the non-zero eigenvalues of ' ', and let v1; : : : ; vr be a corresponding othronormal eigenbasis. For w1; : : : ; wr defined as wi = '(vi)/σi, we have that 1. fw1; : : : ; wrg form an orthonormal set. 2. For all i 2 [r] ∗ '(vi) = σi · wi and ' (wi) = σi · vi : The values σ1; : : : ; σr are known as the (non-zero) singular values of '. For each i 2 [r], the vector vi is known as the right singular vector and wi is known as the right singular vector corresponding to the singular value σi.
    [Show full text]
  • Computing Singular Values of Large Matrices with an Inverse-Free Preconditioned Krylov Subspace Method∗
    Electronic Transactions on Numerical Analysis. ETNA Volume 42, pp. 197-221, 2014. Kent State University Copyright 2014, Kent State University. http://etna.math.kent.edu ISSN 1068-9613. COMPUTING SINGULAR VALUES OF LARGE MATRICES WITH AN INVERSE-FREE PRECONDITIONED KRYLOV SUBSPACE METHOD∗ QIAO LIANG† AND QIANG YE† Dedicated to Lothar Reichel on the occasion of his 60th birthday Abstract. We present an efficient algorithm for computing a few extreme singular values of a large sparse m×n matrix C. Our algorithm is based on reformulating the singular value problem as an eigenvalue problem for CT C. To address the clustering of the singular values, we develop an inverse-free preconditioned Krylov subspace method to accelerate convergence. We consider preconditioning that is based on robust incomplete factorizations, and we discuss various implementation issues. Extensive numerical tests are presented to demonstrate efficiency and robust- ness of the new algorithm. Key words. singular values, inverse-free preconditioned Krylov Subspace Method, preconditioning, incomplete QR factorization, robust incomplete factorization AMS subject classifications. 65F15, 65F08 1. Introduction. Consider the problem of computing a few of the extreme (i.e., largest or smallest) singular values and the corresponding singular vectors of an m n real ma- trix C. For notational convenience, we assume that m n as otherwise we can× consider CT . In addition, most of the discussions here are valid for≥ the case m < n as well with some notational modifications. Let σ1 σ2 σn be the singular values of C. Then nearly all existing numerical methods are≤ based≤···≤ on reformulating the singular value problem as one of the following two symmetric eigenvalue problems: (1.1) σ2 σ2 σ2 are the eigenvalues of CT C 1 ≤ 2 ≤···≤ n or σ σ σ 0= = 0 σ σ σ − n ≤···≤− 2 ≤− 1 ≤ ··· ≤ 1 ≤ 2 ≤···≤ n m−n are the eigenvalues of the augmented matrix| {z } 0 C (1.2) M := .
    [Show full text]
  • The Singular Value Decomposition
    The Singular Value Decomposition Prof. Walter Gander ETH Zurich Decenber 12, 2008 Contents 1 The Singular Value Decomposition 1 2 Applications of the SVD 3 2.1 Condition Numbers . 4 2.2 Normal Equations and Condition . 7 2.3 The Pseudoinverse . 7 2.4 Fundamental Subspaces . 8 2.5 General Solution of the Linear Least Squares Problem . 9 2.6 Fitting Lines . 10 2.7 Fitting Ellipses . 11 3 Fitting Hyperplanes{Collinearity Test 13 4 Total Least Squares 15 5 Bibliography 18 1 The Singular Value Decomposition The singular value decomposition (SVD) of a matrix A is very useful in the context of least squares problems. It also very helpful for analyzing properties of a matrix. With the SVD one x-rays a matrix! Theorem 1.1 (The Singular Value Decomposition, SVD). Let A be an (m×n) matrix with m ≥ n. Then there exist orthogonal matrices U (m × m) and V (n × n) and a diagonal matrix Σ = diag(σ1; : : : ; σn)(m × n) with σ1 ≥ σ2 ≥ ::: ≥ σn ≥ 0, such that A = UΣV T holds. If σr > 0 is the smallest singular value greater than zero then the matrix A has rank r. Definition 1.1 The column vectors of U = [u1;:::; um] are called the left singular vectors and similarly V = [v1;:::; vn] are the right singular vectors. The values σi are called the singular values of A. Proof The 2-norm of A is defined by kAk2 = maxkxk=1 kAxk. Thus there exists a vector x with kxk = 1 such that z = Ax; kzk = kAk2 =: σ: Let y := z=kzk.
    [Show full text]